Interest in anime, especially among otaku, can be a very personal thing, and so it is often easy to consider an insult to your favorite anime an insult to you, and so you come up with all sorts of reasons why the other person might be wrong. The more extreme their response, the more extreme yours potentially can be unless you step back for a moment.
One defensive response that I know I’ve used before is that if doesn’t like your favorite show, it’s because they don’t like whatever it is you like most about the show. Using K-On! as an example, if someone talks about how they think it’s a bad show, an easy response would be, “Oh that’s just because you can’t enjoy the simple things in life,” even if that doesn’t even influence whether or not they like it.
The most dangerous aspect of this line of reasoning is that it’s not like it never applies. There are definitely times when a person hates the show for the exact reason you like it, or that they don’t pay attention to the parts that you pay attention to the most. Because these valid instances exist, it becomes very easy to think of it as a “rule,” and then for another person to see you using it and come out with their own very rigid, extreme opinion opposite yours.
Even if people disagree on fundamental levels, I think it is important for arguments not to get too unnecessarily heated and overly personal (though a little I feel is okay). However, I know that doing so can be very difficult because it only takes one person to start attaching insults to their opinion in order for everyone to be doing so, and then from there it’s all too easy to say that criticism simply should not exist at all, ever when that potentially can stifle discussion. There is definitely a middle ground in terms of discussion, even if it doesn’t necessarily exist for opinions.